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China Mathematical Olympiad 1988 problem4

Source: China Mathematical Olympiad 1988 problem4

November 5, 2013
inequalitiesgeometryarea of a triangleHeron's formulainequalities unsolvedChinan-variable inequality

Problem Statement

(1) Let a,b,ca,b,c be positive real numbers satisfying (a2+b2+c2)2>2(a4+b4+c4)(a^2+b^2+c^2)^2>2(a^4+b^4+c^4). Prove that a,b,ca,b,c can be the lengths of three sides of a triangle respectively. (2) Let a1,a2,…,ana_1,a_2,\dots ,a_n be nn (n>3n>3) positive real numbers satisfying (a12+a22+⋯+an2)2>(n−1)(a14+a24+⋯+an4)(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4). Prove that any three of a1,a2,…,ana_1,a_2,\dots ,a_n can be the lengths of three sides of a triangle respectively.
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